Tensor category of $\mathbb{Z}_2$-orbifold of Heisenberg vertex operator algebra and its applications

Abstract

In this paper, we prove the category of finite length modules for the $\mathbb{Z}_2$-orbifold $M(1)^+$ of the Heisenberg vertex operator algebra whose simple composition factors are $M(1)^\pm$ or $M(1,\lambda)$ for $\lambda \in \mathbb{C}^\times$ is a vertex and braided tensor category. Our strategy is to show these simple composition factors are $C_1$-cofinite and the category of finite length $M(1)^+$-modules is exactly the category of grading-restricted $C_1$-cofinite modules. We also determine the fusion product decompositions of simple objects and prove the rigidity of this category. As an application of the tensor category structure of $M(1)^+$-modules, we prove the category $\mathcal{C}_{-1}(sp(2n))$ of grading-restricted generalized modules for the simple affine vertex algebra $L_{-1}(sp(2n))$ is semisimple. For this, we first prove $M(1)^+$ and simple affine vertex algebra $L_{-\frac{1}{2}}(sp(2n))$ form a commutant pair in the simple minimal $W$-algebra $W_{-1}^{min}(sp(2n))$ for $n \geq 2$ and determine $W_{-1}^{min}(sp(2n))$ as well as its irreducible modules obtained from quantum Hamilton reduction as decompositions of $M(1)^+ \otimes L_{-\frac{1}{2}}(sp(2n))$-modules, then we show all the highest weight modules for $L_{-1}(sp(2n))$ in $\mathcal{C}_{-1}(sp(2n))$ are irreducible via the quantum Hamilton reduction. We also prove a Schur-Weyl duality between $L_{-1}(sp(2n))$ and $M(1)^+$ by showing they form a commutant pair in the $\mathbb{Z}_2$-orbifold of the rank $n$ $\beta\gamma$ system, and then establish a braided reversed equivalence between the category $\mathcal{C}_{-1}(sp(2n))$ and the full subcategory of $C_1$-cofinite $M(1)^+$-modules consisting of direct sums of irreducible modules $M(1)^\pm$ and $M\big(1, \frac{s}{\sqrt{-2n}}\big)$ for $s \in \mathbb{Z}_{\geq 0}$.